 Solution

Use the preceding fixed point with $q=\perp$. If the chosen arithmetic interpretation sends $\perp$ to a PA-refutable sentence $Z$, it gives the arithmetic sentenceThis follows from Arithmetical soundness of GL. Since Peano arithmetic proves $\neg Z$, the two implications are equivalent to negated provability:Thus $C$ expresses the formal consistency statement for Peano arithmetic and is also a negative provability fixed point, like its Gödel sentence $G$.

Apply the supplied modal uniqueness formula to the context with a modalized occurrence $X(p)=(\square p\to \perp )$, with two fresh variables interpreted as $G$ and $C$. A modalized occurrence means that each indicated variable occurrence lies inside a box. Peano arithmetic proves both arithmetic fixed-point equations. The first derivability condition boxes those equations, so the translated uniqueness formula givesOne must not infer $G\leftrightarrow C$ by reflection. Instead, distribution and the provability of the two propositional projections give $\mathrm{PA}⊢BG\leftrightarrow BC$. Negating and using the two fixed-point equations now yieldsThis proves the Equivalence of a Gödel sentence and a consistency statement using the required modal soundness, fixed point and uniqueness results.